5 - Special Types of Linear Programming Problems

Abstract

This chapter describes a number of special types of linear programming problems. These problems arise in transportation systems, in communication systems, in pipeline systems, in electrical networks, in the planning and scheduling of projects, and in many other applications. Each of these problems can be formulated and solved as a linear programming problem. However, because of their special structure these problems can be more effectively solved by other methods that have been specifically developed to handle them. Degeneracy in a transportation problem has the same meaning as it has for a general linear programming problem. That is, a basic variable has value zero. It is possible that while finding an initial basic feasible solution both a supply and a demand constraint are satisfied simultaneously. On the other hand, there may be a tie for choosing a departing variable. In either case at least one basic variable will have value zero. The chapter describes the minimum cost rule for obtaining an initial basic feasible solution to the transportation problem. A number of other methods are available. A desirable starting method is one that will provide a starting solution that is not far from the optimal one in the sense that only a small number of iterations of the transportation algorithm are required.